A269800 Convolution of A000107 and A027852.

0, 0, 1, 3, 10, 30, 91, 268, 790, 2308, 6737, 19609, 57044, 165796, 481823, 1400028, 4068577, 11825459, 34380152, 99981942, 290854486, 846397344, 2463892294, 7174933683, 20900764811, 60904875999, 177535250815, 517673673674, 1509950058629, 4405547856394, 12857716906991

Offset: 0

Author

R. J. Mathar, Mar 05 2016

Keywords

• nonn

Comments

This counts the arrangements of n nested circles in the plane where one pair of circles touches. a(2)=1 because the (only) pair must touch. a(3)=3 because either the third circle circumscribes the touching pair or is inside one of the touching circles or is entirely separated from the touching pair.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO](2016), row sums Table 8.

Crossrefs

Cf. A000107, A027852.

Programs

Mathematica

b[0] = 0; b[1] = 1; b[n_] := b[n] =Sum[Sum[d b[d], {d, Divisors[j]}] b[n - j], {j, 1, n - 1}]/(n - 1);
a7[n_] := a7[n] = b[n] + Sum[ a7[n - i] b[i], {i, 1, n - 1}];
c[n_] := c[n] = If[n <= 1, n, (Sum[Sum[d c[d], {d, Divisors[j]}] c[n - j], {j, 1, n - 1}])/(n - 1)];
a52[n_] := (Sum[c[i] c[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, c[n/2], 0])/2;
a[n_] := Sum[a7[k] a52[n - k + 1], {k, 0, n + 1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 16 2018, after Alois P. Heinz in A000107 and A027852 *)